
\(S\)  and \(S_{ode}\)  perturb to nearby \(C^r-\) smooth locally invariant slow manifolds \(S_{\varepsilon ,h}\)  and \(S_{ode,\varepsilon }\)  respectively, such that
\(\textup {dist}(S_{\varepsilon ,h} , S_{ode,\varepsilon } ) = O(\varepsilon ^2 h) .\) 
For each fixed \(h\)  the manifold \(S_{\varepsilon ,h}\)  has all the properties described in Theorem REF , and the manifold \(S_{ode,\varepsilon }\)  is a Fenichel slow manifold.

\(S_{\varepsilon ,h}\)  and \(S_{ode,\varepsilon }\)  have local graph representations
\(S_{\varepsilon ,h} = \left\lbrace  (x, \varphi _{\varepsilon ,h}(x) ) : x \in \mathcal {K} \right\rbrace  , \qquad S_{ode,\varepsilon } = \left\lbrace  (x, \varphi _{ode,\varepsilon }(x) ) : x \in \mathcal {K} \right\rbrace  ,\) 
where
\(\begin{split}\varphi _{\varepsilon ,h}(x) &= \varphi _0(x) - \varepsilon (D_yf)^{-1} (Df N)^{-1} Df G |_S + O(\varepsilon ^2 h) , \\\varphi _{ode,\varepsilon }(x) &= \varphi _0(x) - \varepsilon (D_yf)^{-1} (Df N)^{-1} Df G |_{S_{ode}} + O(\varepsilon ^2) .\end{split}\) 

\(W^{s/u}_{loc}(S)\)  and \(W^{s/u}_{loc}(S_{ode})\)  perturb to \(C^r-\) smooth stable/unstable manifolds \(W^{s/u}_{loc}(S_{\varepsilon ,h})\)  and \(W^{s/u}_{loc}(S_{\varepsilon , ode})\)  respectively, such that
\(\textup {dist}(W^{s/u}_{loc}(S_{\varepsilon ,h}) , W^{s/u}_{loc}(S_{ode,\varepsilon }) ) = O(\varepsilon ^2 h) .\) 
For each fixed \(h\) , the manifolds \(W^{s/u}_{loc}(S_{\varepsilon ,h})\)  have all the properties described in Theorem REF , while the manifolds \(W^{s/u}_{loc}(S_{ode,\varepsilon })\)  are described by Fenichel theory [1]}, [2]}, [3]}, [4]}.

The manifolds \(W^{s/u}_{loc}(S_{\varepsilon ,h})\)  and \(W^{s/u}_{loc}(S_{\varepsilon , ode})\)  admit foliations by stable/unstable fibers \(w^{s/u}_{loc}(S_{\varepsilon ,h})\)  and \(w^{s/u}_{loc}(S_{\varepsilon , ode})\)  respectively. For each fixed \(h\) , the foliations of \(W^{s/u}_{loc}(S_{\varepsilon ,h})\)  are described by Theorem REF , while the foliations of \(W^{s/u}_{loc}(S_{ode,\varepsilon })\)  are described by Fenichel theory.
Contraction along stable fibers \(w^{s}_{loc}(S_{\varepsilon ,h})\)  occurs with rate \(\chi _A(h)\)  fixed and bounded in the interval
\(1 - \nu _{A,ode} h + c_A h^2 < \chi _A(h) < 1 ,\) 
for a constant \(c_A\) , and repulsion along unstable fibers \(w^{u}_{loc}(S_{\varepsilon ,h})\)  occurs with rate \(\chi _R(h)\)  fixed and satisfying
\(\chi _R(h) > 1 + \nu _{R,ode} h + c_R h^2 > 1 ,\) 
for a constant \(c_R\) .


